Zero-crossings of a wavelet transform

The completeness, stability, and application to pattern recognition of a multiscale representation based on zero-crossings is discussed. An alternative projection algorithm is described that reconstructs a signal from a zero-crossing representation, which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero-crossings. The reconstruction algorithm has a fast convergence and each iteration requires O(N log/sup 2/ (N)) computation for a signal of N samples. The zero-crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems. As an example, the implementation and results of a coarse-to-fine stereo-matching algorithm are described. >

[1]  Yehoshua Y. Zeevi,et al.  Image reconstruction from zero crossings , 1986, IEEE Trans. Acoust. Speech Signal Process..

[2]  Andrew P. Witkin,et al.  Scale-space filtering: A new approach to multi-scale description , 1984, ICASSP.

[3]  Mark J. T. Smith,et al.  Exact reconstruction techniques for tree-structured subband coders , 1986, IEEE Trans. Acoust. Speech Signal Process..

[4]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[5]  Stéphane Mallat,et al.  Multifrequency channel decompositions of images and wavelet models , 1989, IEEE Trans. Acoust. Speech Signal Process..

[6]  W. Eric L. Grimson,et al.  Computational Experiments with a Feature Based Stereo Algorithm , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[7]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[8]  Alan L. Yuille,et al.  Scaling Theorems for Zero Crossings , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[9]  Robert Hummel,et al.  Reconstructions from zero crossings in scale space , 1989, IEEE Trans. Acoust. Speech Signal Process..

[10]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[11]  Tomaso Poggio,et al.  A Theory of Human Stereo Vision , 1977 .

[12]  Shlomo Shamai,et al.  Reconstruction of nonperiodic two-dimensional signals from zero crossings , 1987, IEEE Trans. Acoust. Speech Signal Process..

[13]  B. Logan.  Information in the zero crossings of bandpass signals , 1977, The Bell System Technical Journal.

[14]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[16]  D. Youla,et al.  Image Restoration by the Method of Convex Projections: Part 1ߞTheory , 1982, IEEE Transactions on Medical Imaging.

[17]  T. Poggio,et al.  A computational theory of human stereo vision , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[18]  A. Rosenfeld,et al.  Edge and Curve Detection for Visual Scene Analysis , 1971, IEEE Transactions on Computers.

[19]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .